2013 IEEE International Conference on Robotics and Automation (ICRA)
Karlsruhe, Germany, May 6-10, 2013
Path Planning with Uncertainty: Voronoi Uncertainty Fields
Kyel Ok, Sameer Ansari, Billy Gallagher, William Sica, Frank Dellaert, and Mike Stilman
Abstract— In this paper, a two-level path planning algorithm
that deals with map uncertainty is proposed. The higher
level planner uses modified generalized Voronoi diagrams to
guarantee finding a connected path from the start to the goal
if a collision-free path exists. The lower level planner considers
uncertainty of the observed obstacles in the environment and
assigns repulsive forces based on their distance to the robot
and their positional uncertainty. The attractive forces from the
Voronoi nodes and the repulsive forces from the uncertaintybiased potential fields form a hybrid planner we call Voronoi
Uncertainty Fields (VUF). The proposed planner has two
strong properties: (1) bias against uncertain obstacles, and (2)
completeness. We analytically prove the properties and run
simulations to validate our method in a forest-like environment.
I. I NTRODUCTION
There exist numerous applications that could benefit from a
mobile robot autonomously navigating through a dense environment such as a forest. Wildlife observation, search and
rescue of stranded hikers, environment monitoring, and forest
fire surveillance are some examples. With these applications
in mind, we propose a hierarchical planner for autonomous
navigation and exploration in unknown environments and
simulate a UAV navigating through a forest-like environment.
Autonomous navigation in an environment with densely
distributed obstacles can be achieved using various planning
algorithms. However, many of these algorithms assume an
accurate prior knowledge of the environment and are often
inadequate by themselves for navigating in an unknown
environment. State-of-the-art Simultaneous Localization and
Mapping (SLAM) techniques attempt to overcome this lack
of prior knowledge by gradually building an estimated map
of the surrounding environment [1]. However, noise in sensor
readings introduce growing uncertainty into these maps.
Typical planning algorithms neglect to address this problem of uncertainty in the information provided. Many works
on planning algorithms focus on achieving optimality and
reducing computational complexity and often overlook the
uncertainty of the information being used. Similarly, typical
SLAM algorithms solve localization and mapping problems
and do not consider the later usage of the acquired information. Our algorithm solves this lack of connection between
SLAM and planning by using the uncertainty of the mapped
obstacles in the planning stage to minimize the chance of
collision when navigating. This idea belongs to the family
of algorithms that combines SLAM and planning techniques
together and is commonly denoted as Simultaneous Planning
Localization and Mapping (SPLAM).
† This work was supported in part by NSF grant IIS-1017076.
The authors are with the Center for Robotics and Intelligent Machines,
Georgia Tech, Atlanta, GA, USA. {kyelok, elucidation,
bgallagher, dellaert, mstilman}@gatech.edu
978-1-4673-5642-8/13/$31.00 ©2013 IEEE
Fig. 1. A simulation of 500 obstacles showing the goal (red star), Voronoi
decomposition (thin blue lines), traveled path (thick blue lines), estimated
obstacle positions (green circles), and local potential field (bottom-right) for
one of the iterations. As illustrated, Voronoi decompositions and potential
fields can be combined smoothly with no overhead computations.
Previous works in SPLAM have been geared toward
manipulating the planning algorithms to aid the map building and localization processes in SLAM techniques. For
example, [2] uses frontier points as local destinations to
minimize the system uncertainty and increase the exploration
efficiency. Furthermore, [3] presents belief roadmaps (BRM)
to demonstrate reducing localization uncertainty by selecting
paths with more observable landmarks. Similarly, many
works [4], [5], [6] plan trajectories that result in minimal
uncertainty in its estimates and the maximal coverage area.
Only few works consider the opposite: navigating more
cautiously in regions with high uncertainty instead of attempting to lower the uncertainty of these regions. [7]
uses sensor, localization, and mapping uncertainty in SLAM
algorithms to bias branch extensions in Rapidly-Exploring
Random Trees (RRTs) [8] to nodes with potentially lower
localization uncertainty. However, this method fundamentally
bases on RRTs and inherits the sampling-based planner’s
sub-optimality and probabilistic completeness. Our algorithm
improves on current works by achieving completeness.
We propose a hierarchical planner, Voronoi Uncertainty
Fields (VUF), that guarantees completeness while directly
accounting for the uncertainty of the observed obstacles.
VUF plans a locally best path in terms of shortness and
certainty of no-collision, while navigating to a global goal.
We discuss the details of our algorithm, shown in Figure 1,
in the next section then present simulation results along with
a proof of completeness for the planner. Further discussions
will follow to analyze and denote areas of improvement.
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Fig. 2. Gradient descent method trapped by a local minimum, preventing
from hitting the wall of obstacles (green spheres). The global planner will
pull the robot (magenta) out of this state around the wall if a path exists.
II. VORONOI U NCERTAINTY F IELDS
Voronoi Uncertainty Fields is a hierarchical planner that has a
top-level planner that forms local way-points using Voronoi
vertices and a bottom-level planner that locally refines the
actual path using uncertainty-biased potential fields.
The top-level global planner uses a modified version of
generalized Voronoi diagrams (GVDs) [9] to form a graph
of the collision-free space. By traversing through the nodes in
the graph, if there is a free path from the source to the goal, it
is guaranteed that a path also exists in the graph. Relying on
this property, the global planner creates and updates a list
of Voronoi nodes that forms the shortest path to the goal.
Then, the bottom-level local planner, which fundamentally
bases on potential fields [10], chooses the closest node in the
list and uses it as an attractor, or a local way-point. Based
on the local way-point and the uncertainty-biased repulsive
forces exerted by the nearby obstacles, a sum of forces can
be calculated for the surrounding local area. This resultant
force is then turned into control inputs for the robot.
Although, both Voronoi diagrams and potential fields are
well-studied algorithms in the field of Robotics, our method
of hierarchically combining and manipulating the algorithms
leads to strong properties such as completeness, fast runtime, and overcoming the local minima problem in potential
fields, as illustrated in Figure 2. We exploit the division
of levels to allow more computationally expensive Voronoi
decomposition on the global map to run at a lower frequency
while the cheap local planner runs at a much higher frequency to refine the current path and react to immediate
threats from nearby obstacles.
There are other planners that exploit the idea of combining
multiple planners to compensate for the weakness in each.
[11] uses centroid of Voronoi regions as attractors while
obstacles and boundary exert short range repulsive forces.
Some other planners for mobile robots [12], [13] also use
Voronoi diagrams and potential fields together to overcome
the shortcomings in each planner. Our work builds on this
family of hybrid planners by accounting for the uncertainties
in the map, and further coupling with a SLAM system.
Fig. 3. Potential field contour with densely scattered tree-like obstacles
(green circles). The locally best path for the robot (magenta) at the top-left
corner away from the obstacles to the local goal (cyan) at the bottom is the
gradient descent in the field, as traced in black.
A. SLAM
For the localization and mapping of the robot, we assume a
generic SLAM system is implemented. The vehicle motion
is defined as the following:
Xt+1 = f (Xt , ut ) + ε
(1)
where the robot state X at time t + 1 is only dependent
on the current state Xt , and the control input ut with zero
mean uncorrelated Gaussian noise ε ∼ N (0, Qt ). Similarly,
the measurement is defined as
Z = h (Xt , Lt ) + η
(2)
where a measurement is a function of the robot state X, and
the landmark state L with some Gaussian noise η.
Following a state update using any of the well-known
SLAM algorithms such as EKF-SLAM [14], we are interested in the current uncertainty of the landmarks to bias
against uncertain obstacles in the planning stage after. This
uncertainty in the landmarks is computed by taking the
marginal probability of each landmark as follows:
P (Ltk ) =
Z
t
m
Y
Y
P (X0 ) (Xi | Xi−1 )
P (Lj )P (Zij | Xi , Lj ) dv
v
i=1
j=1
(3)
where v = X1..t , L1..t \ Lk .
The marginal probability of the landmarks is approximated
by a Gaussian distribution P (Ltj ) ≈ N (L¯tj , Σtj ) and the
covariance of this distribution is discriminatively used in the
local planning stage to bias against uncertain obstacles.
B. Local Planner
After calculating the uncertainty of the obstacles, the local
planner takes in a list of the currently visible obstacles and
attempts to find a path from the current position to the
local goal generated by the global planner. It does this by
calculating a potential field and using gradient descent to
form the path, as illustrated in Figure 3.
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The potential field, p, is comprised of three components
as given by Equation 4: 1) the linear distance from the local
goal, dlg , 2) the linear distance from an obstacle, di 3) the
value of a Gaussian distribution, pi , based on its uncertainty,
σi . Component 1 ensures that the potential field will push the
robot towards its intended goal. Component 2 ensures that
the robot won’t get too close to any obstacles, and component
3, given by Equation 5, ensures that uncertain obstacles will
push the robot further away than well localized ones. Each
of these components is empirically weighted by kj to form
a smooth and consistent continuous function. Sensitivity of
these weights do not affect the system significantly, as long
as each component is in reasonable scale to others.
p = k1 dlg + k2 pi + k3 di
1
(4)
d2
− i2
2σ
i
e
(5)
2πσi2
Given the potential field function, gradient descent methods
find the minimum of the function by gradually stepping down
the steepest slope iteratively. This is done by estimating the
gradient, ∇f , of the function at each step, and taking a step in
∇f
the − |∇f
| direction with a step size based on the magnitude
of the gradient. This continues until the gradient goes to zero,
the step size gets tiny, or the robot reaches its goal.
This method, however, can not traverse uphill and will get
trapped by local minima of the potential function. Typically,
this is considered problematic, since the algorithm is unable
to find a path to the goal. In our algorithm, we overcome
the local minima problem by fusing with the global planner.
As shown in Figure 2, when the robot is stuck in a local
minimum, we simply remove the current local goal from the
global planner’s graph, and re-find a new shortest path. Accounting for the sensing uncertainty, we do not immediately
do this but wait for sufficient measurements to be made. If the
robot is detected to be stuck at time s, then the uncertainty
in the pose Xs can be computed by marginializing
pi = p
P (Xs ) =
Z
s
m
Y
Y
P (X0 ) (Xi | Xi−1 )
P (Lj )P (Zij | Xi , Lj ) dv
v
i=1
j=1
(6)
where v = X1..s−1 , L1..t . Then, as the robot stays stationary,
for a future time k s.t. k > s, Equation 3 reduces down to
m
Y
1..m
P (Lk | Zk ) ≈ P (Xs ) P (Lk )
P (Zki | Lk ). (7)
i=1
This marginal probability of obstacle Lk results in more
measurement likelihood P (Zki | Lk ) being multiplied as
new measurements are made. Since these are approximately
Gaussian, by the well-known property, the resulting Gaussian
has a smaller covariance, i.e. over time the uncertainty of the
obstacles decrease. Thus, we use a heuristic to re-plan only
when the uncertainty of the nearby obstacles drops below a
threshold. Note that this approach requires the observation
matrix to be full rank, and this limitation is further discussed
in a later section.
Fig. 4. Global Planner with robot position (purple diamond), global goal
(red star), Voronoi edges (blue dotted lines), observed trees (green circles),
and the shortest path (yellow solid line) are shown. The polygons added to
far-distanced points and the goal, shown at each corner, ensure that Voronoi
edges exist for entry into and out of the graph, and in no-obstacle scenarios.
C. Global Planner
The global planner uses generalized Voronoi diagrams1 to
divide the space into Voronoi regions. Given a set S of n
obstacles on the ground plane, the dominance of obstacle p
over q, where p, q ∈ S, is defined as
dom(p, q) = x ∈ R2 | δ(x, p) ≤ δ(x, q)
(8)
where δ is the euclidean distance function [15]. This dominance is used to decompose the ground plane into regions where each region is dominated by an obstacle. The
graph formed by the boundaries of these Voronoi regions is
searched by the global planner to generate a collision-free
path from the current position to the goal.
However, in order to guarantee completeness and account
for cases where Voronoi decomposition fails, several changes
are made to the original Voronoi graph. First, it is known
that if a collision-free path exists from the start to the goal,
then there exists a collision-free path which traverses only
the generated GVD graph except for the stages entering and
exiting the graph [16]. We address the problem of entering
and exiting the Voronoi graph by always including regular
polygon of Voronoi sites around the robot and the goal to
force Voronoi edges connected to them in the Voronoi graph.
As shown in Figure 4, the edges formed by the polygons
around the goal create multiple connections to the Voronoi
graph in various directions. Also, two far distanced sites are
added in similar fashion to deal with the special case of
Voronoi decomposition failing with zero obstacles in the map
[15]. The Voronoi space is then further processed to prune
incorrect edges. First the Voronoi edges that are too close
to the obstacles are removed from the graph to account for
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1 If
all obstacles are point-like, regular Voronoi diagrams is used instead
Fig. 5. A simulation of 1,000 obstacles showing the Voronoi decomposition
(thin blue), traveled path (thick blue), and estimated obstacle positions
(green). Note that the actual path does not exactly follow the Voronoi edges
but uses them as local way-points.
Fig. 6. Simulations of special cases are shown. Top: robot breaking out
of two local minima, circled in red. Bottom-left: a caged environment with
no solution to the goal. Bottom-right: our complete planner returned no
solution after trying all directions.
the robot size and adds some safety margins to avoid tight
passages. Each edge vi that does not satisfy the criteria
δ(vi , Lk ) < D + τ
(9)
are removed for all obstacles L1..j where δ is the euclidean
distance function, D the robot diameter and τ safety margin.
Then, redundant vertices that have no edges connected to
them or do not connect to the goal are removed to reduce
the graph size. The post-processed graph is searched using
A∗ to find the shortest chain of Voronoi edges from start to
goal, and are are stored as a chain of Voronoi nodes between
these edges. These Voronoi nodes are later accessed by the
the local planner to use as the local goal.
D. Complete Cycle
A complete cycle involves SLAM, local planner, global
planner, and a controller all working in a hierarchical order.
In a typical cycle, the SLAM algorithm would first localize
the robot and measure the nearby obstacles to compute the
Gaussian-approximated marginal probability. Based on the
created map and the estimated robot position, the global
planner would find the Voronoi nodes that form the shortest
path in its graph space. The local planner would then segment
out a small region with nearby obstacles and the local goal
on the global path. Using the Gaussian uncertainties and the
information about the local area, the local planner would
generate a direction most desirable in terms of path shortness
and certainty of obstacle positions. A controller would then
actuate the robot towards desired directions, until either
the local goal is reached or the global planner updates the
local goal. This cycle repeats until the robot is within a
certain distance to the global goal. We simulate this cycle
in MATLAB, using several different scenarios.
III. E XPERIMENTS
We have simulated Voronoi Uncertainty Fields in MATLAB,
in random environments with tree-like obstacles. The start
and goal positions were arbitrary set at a far distance from
each other and a bounding box was made to constrain
the environment. In order to simplify the full navigation
problem to fix the scope on the planner, we abstract out
the measurement, SLAM and controls problems. For the
measurement, the observation matrix is assumed to be fullrank where the information about the obstacles nearby are
obtained each iteration with uncertainty decreasing with
more measurements. At each iteration, we consider only the
uncertainty in the landmarks in the current local frame, and
assume that a graph of such local frames can be optimized as
done in relative SLAM works such as [17]. Finally, perfect
actuation is assumed to abstract out the controls problem.
We simulated different scenarios with varying numbers
and positions of obstacles. The obstacle number ranged from
0 to 10,000 obstacles and the positions were either random
placements or normal distributions. Some tests for special
cases were also performed to verify our planner’s ability to
break out of local minima, and terminate if no solution exists.
1) 1,000 Trees: In Figure 5, a 1,000 obstacle scenario is
shown with only the information known to the planner. Other
than the surrounding area en route to the goal, the rest of
the area out of the sensor range is not revealed.
2) Escaping Local Minima: Another scenario simulated is
to verify the planner’s ability to escape local minima. Shown
in Figure 6, there are two local minima that temporarily
block the robot. The global planner, which keeps track of
last known positions, is able to choose a new local goal that
navigates the robot out of the local minima.
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Fig. 8. Timing graph showing the time taken for the local planner, the
global planner (not accounting for the pruning time), the pruning time,
and the total combined time. It can be observed that most overhead comes
from the inefficient pruning stage while both planners are near real-time
performance without any code optimizations.
Fig. 7. Stress testing the planner with 10,000 obstacles in the environment.
The planner takes a long detour but eventually arrives at the goal.
3) No Solutions: To verify the completeness of our planner, a few scenarios with no solutions were also simulated.
Shown in Figure 6, when the planner was run in an environment consisting of a robot boxed in a square of obstacles
with the goal outside the box, the planner continued to search
for a path until uncertainty of the obstacles dropped below
a threshold, where the planner terminated with no solution.
4) 10,000 Trees Stress Test: A large scenario with 10,000
normally distributed obstacles was used to test the planner’s
ability to utilize the global planner’s ability to back-track and
find new paths when detecting dead ends. As shown in Figure
7, obstacles were normally distributed at an offset from the
center of the map, and the robot was started in the bottomleft corner with the goal in the top-right corner of the field.
As seen in the path, the robot initially travels directly towards
the goal, but upon encountering obstacles it navigates to the
right counter-clockwise searching for a path. At a certain
point at the bottom of the mass of obstacles, it determines
that the cost of traveling any further counter-clockwise is
more than it is to back-track and navigate clockwise, which
it then does and successfully reaches the goal.
In our simulations, we have demonstrated that (1) the robot
is able to back-track and reach the goal, (2) escape local
minima, and (3) return with no solution if there is no path.
In the following section, we analyze and discuss the observed
properties and prove that our planner is complete.
IV. A NALYSIS
A. Completeness
We define our workspace W as a bounded 2-dimensional
space (i.e. ground plane for a ground robot). The workspace
can be Voronoi decomposed into a configuration space V
of Voronoi regions, V = [v1 , ...vn ], using Equation 8. If all
sites, or obstacles in the workspace, are not collinear, each
Voronoi region vi has a connected boundary and the Voronoi
diagram is connected, as proven in Theorem 7.4 [18].
Then, traversability from one Voronoi region vp to a
neighboring Voronoi region vq , across one segment of its
connected boundary (commonly termed Voronoi edge) can
be calculated using Equation 9. Our configuration space V
of finite number of regions v1..n can then be represented as a
graph where the nodes are Voronoi edges of these connected
regions and a graph edge between two nodes is traversability
between the two regions. It is well known that A* search on
such graph for a path from one node vp to to another vq is
complete [19]. With pre-processing to remove collinearity,
our global Voronoi planner that uses this search to traverse
from current Voronoi region to the goal region along the
boundary edges is then complete.
B. Optimality
Contrary to completeness, optimality is not guaranteed for
the global planner. Since the representation of the world
is constantly updating as the robot progresses, it is not
possible to have a globally optimal solution before all regions
have been explored. Although A* search with admissible
euclidean distance heuristic used in the global planner does
obtain the shortest path to the goal, it is only the shortest
distance in the Voronoi graph, and may not be the shortest
path in the region.
C. Computation Time
Timing tests were done in a uniform environment to explore
the possibilities of real-time operation of the planner. Table
I shows the time taken on average for a single iteration of
the global and the local planner in a map of 200 obstacles
with pruning of the Voronoi graph turned on and off.
While the local planner is fast and only uses a local
subset of obstacles, the global planner uses every obstacle
observed and loses scalability. For example, navigating in
the map of 200 obstacles, the local planner only interacted
with 34.4 obstacles on average, whereas the global planner
interacted with 139 obstacles in the last ieration. Despite
the scalability issue, both the global and the local planners
are well manageable for real-time operations as evidenced in
Figure 8 as long as the pruning of the Voronoi graph is turned
off. Efficient implementation of the pruning stage remains as
an area of improvement for real-time operations.
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TABLE I
T IME TAKEN FOR A SINGLE ITERATION AT THE BEGINNING AND THE
END OF THE OPERATION FOR BOTH LOCAL AND GLOBAL PLANNERS .
Iteration
First
Last
Iteration
First
Last
200 obstacles (without pruning)
Total (s) Global Plan (s) Local Plan (s)
0.528
0.0416
0.0822
1.026
0.590
0.0572
200 obstacles (with pruning)
Total (s) Global Plan (s) Local Plan (s)
0.333
0.051
0.022
12.233
10.477
1.025
V. D ISCUSSION AND F UTURE W ORK
Voronoi Uncertainty Fields (VUF) is a new deterministic,
complete collision-free algorithm for navigation in uncertain
environments. Results in Figure 7 show that the global map
generated during the planner’s operation can be used to replan when a dead end is encountered. In Figure 6, it is
evidenced that the global Voronoi planner can move the robot
out of local minima and correctly terminate when a collisionfree path to the goal does not exist.
Although the simulations demonstrate our planner’s capabilities in ideal situations, extending the planner to use
on an actual mobile robot requires solving the associated
observation, map/pose optimization, and control problems
that were abstracted out in the simulation of the proposed
planner. For example, current observation model assumes a
hypothetical sensor with a full-rank observation matrix and
may pose a problem in realistic implementations. However,
this assumption is only used to reject false-positive local
minima when stuck, and may not be needed with a different
strategy to deal with local minima, i.e. breaking out immediately when stuck and exploring the nearby area or re-visiting
only after no other solutions exist. In addition, the limitation
in the system that the measurement noise must be Gaussian
may need to be overcome for realistic low-cost sensors.
While the controls problem is platform specific and the
current algorithm does not impose any constraints, it would
be desirable to factor in the velocity of the vehicle in the
planner to allow only the paths that can be executed.
It is also possible to extend current algorithm in 2dimensional space to higher dimensions by replacing the
global planner with a high-dimensional planner such as the
RRTs [8]. As long as the replacement global planner can find
a feasible path to the goal and iteratively provide a local goal
to the local planner, i.e. providing the nearest tree node on
a viable path to the goal when replacing with the RRTs, the
local planner would still take account of the uncertainty in the
obstacles and refine the local path. Moreover, such choice of
global planner could also extend uncertainty to be handled in
the global level as well by using techniques such as biasing
node sampling away from uncertain regions [7]. However, a
drawback of replacing the global planner is that the overall
system may lose the the properties of the Voronoi planner,
such as completeness if the replacement planner does not
have such properties.
Looking at the timing results of the simulations in Table
I, our planner could potentially operate in real-time with a
large number of obstacles, once some improvements in the
pruning stage is made and the scalability issue is dealt with.
To gain scalability, a solution that allows the global planner
to limit the number of obstacles it processes each step while
preserving the capability to backtrack upon encountering a
dead-end would be desirable. One possible solution would
be to further divide the hierarchical approach and have the
global planner use a subset of obstacles, and only use the
full global map when no solution exists in the local path.
Although some areas of improvement remain as future
work, we have proposed a novel planner, Voronoi Uncertainty Fields, that deals with map uncertainties in a deterministic and complete way.
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